Do all my dependent variables have to be normal for a non-parametric assumption?

Question

Im currently attacking SPSS, and I'm just testing for normality... In the Shapiro-Wilk
Statistic test, my dependent variables have a wide range of significance levels. Obviously, I know that the sig. should be >0.05, which most the variables are...apart from 1 variable.

Does this mean all of my data is non-parametric and I have to use another test?

• Here is my data below: 0.077, 0.400, 0.169, 0.157.

In bold is the variable which is not normal

Answer 1

As the other Nick (@NickCox) noted, $.077>.05$. Still, I suppose this puts you close enough for concern to the conventional $\alpha=.05$ – that is, I support your intuition to not ignore the result just because it doesn't quite fall below the presumptive threshold, which is quite often a poor way to perform hypothesis testing anyway. As @Glen_b commented on @Germaniawerks' answer, you could fail to achieve significance due to wimpy (as opposed to powerful) sample sizing despite nonnormality. Observe (in R)! shapiro.test(1:46): $W=.96, p=.079$, much like your result.

Compare to superior ways of judging normality:

As the red line shows, 1:46 $(1,2,3...46)$ is very much not normally distributed – it's totally flat – but it still takes 46 observations to achieve even your "insignificant" result with a Shapiro–Wilk test! Well-behaved for small datasets indeed...BTW, see how much more this actually tells you about your data than just "approximately normal" or not? Don't underestimate the value of simply plotting data! Your eyes can often tell you a lot more than some test statistic.

If this doesn't destroy your faith in the Shapiro–Wilk test, have a look at this question: "Is normality testing 'essentially useless'?" It will help you explore these graphical approaches to normality assessment better, and give you a nice intro to the surprising degree of flexibility and conditionality in those parametric assumptions you know and fear. Sometimes the analysis you really want to justify running by performing this test won't actually be all that sensitive to normality violations (hence Glen_b's comment on your OP, I assume). Then again, sometimes it will...

So about using a nonparametric test: just go for it. If it costs you a little power or makes your hypothesis harder to understand, hopefully that just means you have a little more legwork to do in the name of science. If you really can't afford to lose power, then you may prefer a parametric test, but you probably can't afford to use an $\alpha$ threshold for judging the value of the null. If you really do have to make a dichotomous reject/fail to reject decision about the null, you have my sympathy...but I doubt that this is actually necessary either. Beware conventional practices...they can be misleading.

BTW, if you're hesitating to dive into the slightly deeper end of nonparametric analysis, here's another statistical specter to scare you straight: multivariate normality! Depending on what you really want to do with those four variables, you might have to contend with an assumption like that too.

Answer 2

• Non-parametric refers to a test that makes no assumption about the parameters of the data, whereby the parameters of a probability distribution are usually meant.
• The output of the Shapiro-Wilk test is whether your data are more specifically normally distributed.
• The S-W test is well behaved for small to moderate datasets. If you get result of a p-value of 0.07, you cannot reject normality. So, you can use parametric tests for all your data.
• PS: Even if your data were somewhat non-normal you could still apply some of the parametric tests, because of their robustness to departures from normality, such as t-tests and ANOVA, where the Central Limit Theorem kicks in very effectively.